What is the difference between LSTM and fully connected LSTM?
Based on the citations in the ConvLSTM paper I have come to the conclusion that they mean the Peephole LSTM when they say fully connected LSTM. In the paper that they have taken the encoder-decoder-predictor model from, where they refer to a fully connected LSTM, a Peephole LSTM is used. Also they take their fully connected LSTM definition from this paper, which again uses the Peephole LSTM.  

With that the difference would be the added "peephole connections", that lets the gate layers look at the cell states and access the constant error carousel.